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How does air pressure affect the speed of sound in relation to its kinetic theory etc?

I have tried searching this but i have not found a suitable answer as other websites simply related to air pressure waves.

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To first order, the speed of sound is not affected by pressure. Pressure waves can be shown to fulfill the D'Alembert wave equation $(c_S^2\,\nabla^2 - \partial_t^2)\psi=0$ where the wavespeed $c_S$ is given by:

$$c_S = \sqrt{\frac{K}{\rho}}$$

where $K$ is the bulk modulus of the medium in question and $\rho$ its density. Now, for an ideal gas, the bulk modulus $K$ is in most conditions proportional to the pressure; if the compression is adiabatic (good approximation for high frequency sound, as there is little time for heat to shuttle back and forth in the gas), then $K=\gamma\,P$, where $\gamma$ is the Heat Capacity Ratio or Adiabatic Index. However, from the ideal gas law $P\,V=n\,R\,T$ we have:

$$\rho = \frac{n\,M}{V} = \frac{P\,M}{R\,T}$$

where $M$ is the mean molar mass of the gas in question in kilograms. Thus the pressures cancel out in the speed of sound:

$$c_S = \sqrt{\frac{\gamma\,R\,T}{M}}$$

Thus we see that the speed is also weakly affected by the humidity - more water in the air raises the mean molecular mass. If we put $M=0.029$, $T=300K$ and $\gamma = 1.4$ for air, we get $c_S=347{\rm m\,s^{-1}}$.

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